AXIOMS WHICH IMPLY GCH (A THIRD REMARK)
The axioms B and C proposed on FOM on June 22 and June 24 are
equivalent to each other.
This follows immediately from the following theorem.
THEOREM. For any cardinal alpha the sets
P(alpha) intersect OD
and
{Th((V_beta, membership, xi)_(xi < alpha)) : beta in Ord}
are of the same cardinality.
The proof is an easy extension of the (easy) proof for the case
alpha = omega, which is given on page 275 of the paper
J. Myhill and D. Scott, "Ordinal definability", Proc. Sympos. Pure
Math. Part 1, Vol XIII, AMS, 1971.
The only additional observation needed in that extended proof is that the
second set of the above theorem is of cardinality not less than alpha.
Jan Mycielski